The Research Of Some Smarandache Functions Mean Value Problems

Arithmetical function is one of the important researching field of number theory. And the mean value of arithmetical functions is the problem that many number theory researchers like to tackle with. So, in this thsis, we make some elementary research on the Smarandache arithmetical functions’mean value.First, we research the r-th root sequence. We look for some functions which can be compounded or mixed with the r-th root sequence, and use elementary or analyzing methods to research the mean value on the composite and blended functions to get some asymptotic formulas.Then, we discuss the questions about structured sequences in the book of 《Only Problem, Not Solutions》 that written by American-Romanian number theorist Florentin Smarandache. Let f(n) is the sum of each digit of the n-th item in this sequence, and let g(n) is the n-th item. We use elementary methods to research the mean value of f(n) and g(n), and get some interesting asymptotic formulas.Finally, we research the2n-digital sequence. In similar, we use elementary and analyzing methods to research the mean value of the composite functions to get some asymptotic formulas.